Only a few open clusters are as important for the study of stellar and substellar objects, and their formation and evolution, as the young sigma Orionis cluster. However, a complete spectroscopic characterisation of its whole stellar population is still missing. We filled most of that gap with a large spectroscopic and astrometric survey of targets towards sigma Orionis. Eventually, it will be one of the open clusters with the lowest proportion of interlopers and the largest of confirmed cluster members with known uncontrovertible youth features. We acquired 317 low-resolution optical spectra with IDS at the 2.5m Isaac Newton Telescope and OSIRIS at the 10.4m Gran Telescopio Canarias. On them, we measured equivalent widths of LiI, H{alpha}, and other key lines, and determined spectral types. We complemented the information with Gaia DR2 astrometric data and other features of youth (mid-infrared excess, X-ray emission) compiled with Virtual Observatory tools and from the literature. Of the 168 observed targets, we determined for the first time spectral types of 39 stars and equivalent widths of LiI and H{alpha} of 34 and 12 stars, respectively. We identified 11 close ({rho}<~3-arcsec) binaries resolved by Gaia, of which three are new, 14 strong accretors, of which four are new and another four have H{alpha} emission shifted by over 120km/s, two juvenile star candidates in the sparse population of the Ori OB1b association, and one spectroscopic binary candidate. Remarkably, we found 51 cluster non-members, 35 of which were previously considered as sigma Orionis members and taken into account in high-impact works on, e.g., disc frequency and initial mass function.